# Without using pen and paper, determine which of the following homogeneous systems have nontrivial solutions

This is another question from Anton, Elementary Linear Algebra 9th Ed. The first system is:

$\begin{array}{1}x_{1}+3x_{2}-x_{3}=0\\ \,\,\qquad x_{2}-8x_{3}=0\\ \,\,\,\,\qquad\qquad 4x_{3}=0\end{array}$

Correction as demonstrated in the comments below. This system does not have a nontrivial solution.

The second system is:

$\begin{array}{1}a_{11}x_{1}+a_{12}x_{2}+a_{13}x_{3}=0\\ a_{21}x_{1}+a_{22}x_{2}+a_{23}x_{3}=0\end{array}$

For this second system it is an underdetermined homogeneous system of linear equations and so it has infinite solutions.

The third system is:

$\begin{array}{1}3x_{1}-2x_{2}=0\\ 6x_{1}-4x_{2}=0\end{array}$

This system is dependent linear equations, so they are collinear with infinite solutions.

The fourth system is:

$\begin{array}{1}2x_{1}-3x_{2}+4x_{3}-x_{4}=0\\ 7x_{1}+x_{2}-8x_{3}+9x_{4}=0\\ 2x_{1}+8x_{2}+x_{3}-x_{4}=0\end{array}$

With this fourth one, I cannot see any obvious sign to tell me there is more than the trivial solution. Is there something here that I am missing. All I can do at a glance is identify that it is not any of the other three cases. Maybe row-reduction might discover something else but this question is not supposed to use pencil and paper, i.e. no computation.

Thanks.

• Surely, in your first system, $x_3$ must be zero? – Lord Shark the Unknown Jun 11 '17 at 6:27
• oops, yes of course, sorry about that. Let me look at that one more closely. – Bucephalus Jun 11 '17 at 6:27
• Ok thanks @LordSharktheUnknown. I can see that this one has only the trivial solution. My bad. Will correct the question. – Bucephalus Jun 11 '17 at 6:29
• The fourth and second systems are of the same type. – amd Jun 11 '17 at 7:19
• Thanks for that @amd. – Bucephalus Jun 11 '17 at 7:23

If you are given m homogeneous equations in n unknowns and $m < n$, then you always get infinitely many solutions, since at least one variable becomes free.
For a homogeneous system, if you have m equations in n unknowns with $m = n$, then you get only the trivial solution.